Invariants
| Level: | $6$ | $\SL_2$-level: | $6$ | Newform level: | $36$ | ||
| Index: | $12$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $1 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (none of which are rational) | Cusp widths | $6^{2}$ | Cusp orbits | $2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 6B1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 6.12.1.4 |
Level structure
Jacobian
| Conductor: | $2^{2}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 36.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ - 18 x w + 3 y^{2} + z^{2} + z w + w^{2} $ |
| $=$ | $36 x^{2} + 3 x w - y^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 10 x^{4} - 2 x^{3} y + x^{2} y^{2} - 11 x^{2} z^{2} + 2 x y z^{2} + 4 z^{4} $ |
Rational points
This modular curve has no real points, and therefore no rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{6}z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{6}y$ |
Maps to other modular curves
$j$-invariant map of degree 12 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^4\cdot3^4\,\frac{w^{3}}{12xz^{2}+12xzw+3xw^{2}+z^{2}w+zw^{2}+w^{3}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_{\mathrm{ns}}(2)$ | $2$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
| $X_{\mathrm{ns}}(3)$ | $3$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_{\mathrm{ns}}(3)$ | $3$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| $X_{\mathrm{ns}}^+(6)$ | $6$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 6.6.1.a.1 | $6$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 6.36.1.a.1 | $6$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
| $X_{\mathrm{ns}}(12)$ | $12$ | $4$ | $4$ | $3$ | $0$ | $1^{2}$ |
| 18.36.1.a.1 | $18$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
| 18.36.3.a.1 | $18$ | $3$ | $3$ | $3$ | $0$ | $2$ |
| 18.36.3.b.1 | $18$ | $3$ | $3$ | $3$ | $0$ | $1^{2}$ |
| 18.36.3.c.1 | $18$ | $3$ | $3$ | $3$ | $0$ | $1^{2}$ |
| 18.36.3.d.1 | $18$ | $3$ | $3$ | $3$ | $0$ | $2$ |
| $X_{\mathrm{ns}}(18)$ | $18$ | $9$ | $9$ | $7$ | $1$ | $1^{4}\cdot2$ |
| 30.60.5.a.1 | $30$ | $5$ | $5$ | $5$ | $2$ | $1^{4}$ |
| 30.72.5.c.1 | $30$ | $6$ | $6$ | $5$ | $0$ | $1^{4}$ |
| 30.120.9.g.1 | $30$ | $10$ | $10$ | $9$ | $3$ | $1^{8}$ |
| 42.36.1.c.1 | $42$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
| 42.96.7.a.1 | $42$ | $8$ | $8$ | $7$ | $1$ | $1^{4}\cdot2$ |
| 42.252.19.e.1 | $42$ | $21$ | $21$ | $19$ | $5$ | $1^{6}\cdot2^{4}\cdot4$ |
| 42.336.25.e.1 | $42$ | $28$ | $28$ | $25$ | $6$ | $1^{10}\cdot2^{5}\cdot4$ |
| 66.144.11.a.1 | $66$ | $12$ | $12$ | $11$ | $2$ | $1^{10}$ |
| 66.660.51.e.1 | $66$ | $55$ | $55$ | $51$ | $19$ | $1^{8}\cdot2^{13}\cdot4^{4}$ |
| 66.660.51.i.1 | $66$ | $55$ | $55$ | $51$ | $19$ | $1^{2}\cdot2^{16}\cdot4^{4}$ |
| 66.792.61.e.1 | $66$ | $66$ | $66$ | $61$ | $21$ | $1^{12}\cdot2^{16}\cdot4^{4}$ |
| 78.36.1.c.1 | $78$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 78.168.13.c.1 | $78$ | $14$ | $14$ | $13$ | $?$ | not computed |
| 102.216.17.a.1 | $102$ | $18$ | $18$ | $17$ | $?$ | not computed |
| 114.36.1.c.1 | $114$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 114.240.19.a.1 | $114$ | $20$ | $20$ | $19$ | $?$ | not computed |
| 126.36.1.c.1 | $126$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 126.36.1.d.1 | $126$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 126.36.3.a.1 | $126$ | $3$ | $3$ | $3$ | $?$ | not computed |
| 126.36.3.b.1 | $126$ | $3$ | $3$ | $3$ | $?$ | not computed |
| 126.36.3.c.1 | $126$ | $3$ | $3$ | $3$ | $?$ | not computed |
| 126.36.3.d.1 | $126$ | $3$ | $3$ | $3$ | $?$ | not computed |
| 126.36.3.e.1 | $126$ | $3$ | $3$ | $3$ | $?$ | not computed |
| 126.36.3.f.1 | $126$ | $3$ | $3$ | $3$ | $?$ | not computed |
| 138.288.23.a.1 | $138$ | $24$ | $24$ | $23$ | $?$ | not computed |
| 186.36.1.c.1 | $186$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 222.36.1.c.1 | $222$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 234.36.1.c.1 | $234$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 234.36.1.d.1 | $234$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 234.36.3.a.1 | $234$ | $3$ | $3$ | $3$ | $?$ | not computed |
| 234.36.3.b.1 | $234$ | $3$ | $3$ | $3$ | $?$ | not computed |
| 234.36.3.c.1 | $234$ | $3$ | $3$ | $3$ | $?$ | not computed |
| 234.36.3.d.1 | $234$ | $3$ | $3$ | $3$ | $?$ | not computed |
| 234.36.3.e.1 | $234$ | $3$ | $3$ | $3$ | $?$ | not computed |
| 234.36.3.f.1 | $234$ | $3$ | $3$ | $3$ | $?$ | not computed |
| 258.36.1.c.1 | $258$ | $3$ | $3$ | $1$ | $?$ | dimension zero |